On the Construction of Recursively Differentiable Quasigroups and an Example of a Recursive $[4,2,3]_{26}$-Code
This work resolves a specific open case in coding theory, contributing incrementally to the understanding of recursive MDS codes and quasigroups.
The paper tackles the problem of constructing recursive MDS codes of dimension 2 and length 4 over finite alphabets, specifically settling the outstanding case for alphabet size q=26, which was previously unresolved. It achieves this by providing an explicit construction using recursively differentiable quasigroups and perfect cyclic Mendelsohn designs, thereby verifying the conjecture for all q except 2 and 6.
In 1998, E. Couselo, S. González, V. T. Markov, and A. A. Nechaev introduced the notions of recursive codes and recursively differentiable quasigroups. They conjectured that recursive MDS codes of dimension $2$ and length $4$ exist over every finite alphabet of size $q \not\in \{2, 6\}$, and verified this conjecture in all cases except $q \in \{14, 18, 26, 42\}$. In 2008, V. T. Markov, A. A. Nechaev, S. S. Skazhenik, and E. O. Tveritinov resolved the case $q=42$ by providing an explicit construction. The present paper settles the outstanding case $q=26$. The construction rests upon methods for producing recursively differentiable quasigroups and recursive MDS codes via perfect cyclic Mendelsohn designs. Moreover, we sharpen several known bounds concerning the existence of recursively $n$-differentiable quasigroups of small orders.